The "Sent folder" for an account defaults to the local folders. However, you can change that:

Menu button -> Configuration

Click on the account

Click on the drop-down at the top of the account configuration screen, and select "Folder settings"

On the screen that comes up, you'll see click-able items where you can change the Sent and Drafts folders used by that account

There is no feature in IMAP to actually tell me which folders to pick for this, so its kind of hard to do it automatically. (Though Google appears to have proposed one, which I think is used by Gmail.) However, I know this could eventually become part of the setup wizard. (Right now, the wizard doesn't configure anything that requires LogicMail to actually add the account configuration and use it.)

Moving/copying is possible, with some caveats:

There is a known (though I should write a ticket) issue with Local folders, where you need to actually open the message before it will let you copy it to other accounts.

Since "Move" is basically a convenient way of doing Copy+Delete, I only support it for cases where its completely safe. Or in other words, in cases where LogicMail doesn't actually have to load/save the raw message data with complete accuracy to do it. So, I only support that for the following cases:

Between local folders (since its just copying a file)

Between IMAP folders (since IMAP has a command to do this server-side)

I probably could support "Move" in a few more cases, but it seemed much easier/safer to just limit it to "Copy" for those.